Question: The lifespans of sloths in a particular zoo are normally distributed. The average sloth lives $14.9$ years; the standard deviation is $1.5$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a sloth living less than $10.4$ years.
Solution: $14.9$ $13.4$ $16.4$ $11.9$ $17.9$ $10.4$ $19.4$ $99.7\%$ $0.15\%$ $0.15\%$ We know the lifespans are normally distributed with an average lifespan of $14.9$ years. We know the standard deviation is $1.5$ years, so one standard deviation below the mean is $13.4$ years and one standard deviation above the mean is $16.4$ years. Two standard deviations below the mean is $11.9$ years and two standard deviations above the mean is $17.9$ years. Three standard deviations below the mean is $10.4$ years and three standard deviations above the mean is $19.4$ years. We are interested in the probability of a sloth living less than $10.4$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $99.7\%$ of the sloths will have lifespans within 3 standard deviations of the average lifespan. The remaining $0.3\%$ of the sloths will have lifespans that fall outside the shaded area. Because the normal distribution is symmetrical, half $({0.15\%})$ will live less than $10.4$ years and the other half $({0.15\%})$ will live longer than $19.4$ years. The probability of a particular sloth living less than $10.4$ years is ${0.15\%}$.